Theorem elcncf1di 14758

Description: Membership in the set of continuous complex functions from A to B. (Contributed by Paul Chapman, 26-Nov-2007.)

Hypotheses

Ref	Expression
elcncf1d.1	|- ( ph -> F : A --> B )
elcncf1d.2	|- ( ph -> ( ( x e. A /\ y e. RR+ ) -> Z e. RR+ ) )
elcncf1d.3	|- ( ph -> ( ( ( x e. A /\ w e. A ) /\ y e. RR+ ) -> ( ( abs ` ( x - w ) ) < Z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) ) )

Assertion

Ref	Expression
elcncf1di	|- ( ph -> ( ( A C_ CC /\ B C_ CC ) -> F e. ( A -cn-> B ) ) )

Distinct variable groups:   x, w, y, A   w, B, x, y   w, F, x, y   ph, w, x, y   w, Z

Allowed substitution hints:   Z( x, y)

Proof of Theorem elcncf1di

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elcncf1d.1	. . 3 |- ( ph -> F : A --> B )
2		elcncf1d.2	. . . . . 6 |- ( ph -> ( ( x e. A /\ y e. RR+ ) -> Z e. RR+ ) )
3	2	imp 124	. . . . 5 |- ( ( ph /\ ( x e. A /\ y e. RR+ ) ) -> Z e. RR+ )
4		an32 562	. . . . . . . . 9 |- ( ( ( x e. A /\ w e. A ) /\ y e. RR+ ) <-> ( ( x e. A /\ y e. RR+ ) /\ w e. A ) )
5	4	anbi2i 457	. . . . . . . 8 |- ( ( ph /\ ( ( x e. A /\ w e. A ) /\ y e. RR+ ) ) <-> ( ph /\ ( ( x e. A /\ y e. RR+ ) /\ w e. A ) ) )
6		anass 401	. . . . . . . 8 |- ( ( ( ph /\ ( x e. A /\ y e. RR+ ) ) /\ w e. A ) <-> ( ph /\ ( ( x e. A /\ y e. RR+ ) /\ w e. A ) ) )
7	5, 6	bitr4i 187	. . . . . . 7 |- ( ( ph /\ ( ( x e. A /\ w e. A ) /\ y e. RR+ ) ) <-> ( ( ph /\ ( x e. A /\ y e. RR+ ) ) /\ w e. A ) )
8		elcncf1d.3	. . . . . . . 8 |- ( ph -> ( ( ( x e. A /\ w e. A ) /\ y e. RR+ ) -> ( ( abs ` ( x - w ) ) < Z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) ) )
9	8	imp 124	. . . . . . 7 |- ( ( ph /\ ( ( x e. A /\ w e. A ) /\ y e. RR+ ) ) -> ( ( abs ` ( x - w ) ) < Z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) )
10	7, 9	sylbir 135	. . . . . 6 |- ( ( ( ph /\ ( x e. A /\ y e. RR+ ) ) /\ w e. A ) -> ( ( abs ` ( x - w ) ) < Z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) )
11	10	ralrimiva 2567	. . . . 5 |- ( ( ph /\ ( x e. A /\ y e. RR+ ) ) -> A. w e. A ( ( abs ` ( x - w ) ) < Z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) )
12		breq2 4034	. . . . . 6 |- ( z = Z -> ( ( abs ` ( x - w ) ) < z <-> ( abs ` ( x - w ) ) < Z ) )
13	12	rspceaimv 2873	. . . . 5 |- ( ( Z e. RR+ /\ A. w e. A ( ( abs ` ( x - w ) ) < Z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) ) -> E. z e. RR+ A. w e. A ( ( abs ` ( x - w ) ) < z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) )
14	3, 11, 13	syl2anc 411	. . . 4 |- ( ( ph /\ ( x e. A /\ y e. RR+ ) ) -> E. z e. RR+ A. w e. A ( ( abs ` ( x - w ) ) < z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) )
15	14	ralrimivva 2576	. . 3 |- ( ph -> A. x e. A A. y e. RR+ E. z e. RR+ A. w e. A ( ( abs ` ( x - w ) ) < z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) )
16	1, 15	jca 306	. 2 |- ( ph -> ( F : A --> B /\ A. x e. A A. y e. RR+ E. z e. RR+ A. w e. A ( ( abs ` ( x - w ) ) < z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) ) )
17		elcncf 14752	. 2 |- ( ( A C_ CC /\ B C_ CC ) -> ( F e. ( A -cn-> B ) <-> ( F : A --> B /\ A. x e. A A. y e. RR+ E. z e. RR+ A. w e. A ( ( abs ` ( x - w ) ) < z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) ) ) )
18	16, 17	syl5ibrcom 157	1 |- ( ph -> ( ( A C_ CC /\ B C_ CC ) -> F e. ( A -cn-> B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2164   A.wral 2472   E.wrex 2473   C_ wss 3154   class class class wbr 4030   -->wf 5251   ` cfv 5255  (class class class)co 5919   CCcc 7872   < clt 8056   - cmin 8192   RR+crp 9722   abscabs 11144   -cn->ccncf 14749

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-map 6706  df-cncf 14750

This theorem is referenced by:  elcncf1ii  14759  cncfmptc  14775  cncfmptid  14776  addccncf  14779  negcncf  14784